where the minimization is taken over all \(\sigma (Y)\) -measurable random variables \(Z.\)  It is well-known that \(\mathbb {E}[X \mid Y]\)  is linear (i.e., a first degree polynomial in \(Y\) ) if and only if \(X\)  is Gaussian (see, e.g., [1]}). We take this a step further and examine when \(\mathbb {E}[X\mid Y]\)  is close to being a polynomial. Specifically, we focus on two questions:
